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Optimal coating thickness for enhancement of optical effects in optical multilayer-based metrologies
Optics Communications 403 (2017) 150–154
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Optimal coating thickness for enhancement of optical effects in optical multilayer-based metrologies Chu Manh Hoang a , Takuya Iida b,c , Le Tri Dat d , Ho Thanh Huy d , Nguyen Duy Vy e,f, * a
International Training Institute for Materials Science, Hanoi University of Science and Technology, No. 1, Dai Co Viet, Hai Ba Trung, Hanoi, Viet Nam Department of Physical Science, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan c Research Institute for Light-induced Acceleration System (RILACS), Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan d Faculty of Physics and Engineering Physics, University of Science, Ho Chi Minh City, Viet Nam e Theoretical Physics Research Group, Ton Duc Thang University, Ho Chi Minh City, Viet Nam f Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b
a r t i c l e
i n f o
Keywords: Fiber-top cantilever Optical multilayer Atomic force microscope Radiation pressure
a b s t r a c t We theoretically determine an optimal configuration for a planar optical multilayer that is used to enhance the radiation pressure (RP) and the thermal heating in fiber-based metrology systems such as fiber-top cantilevers. The RP exerting on the metallic layer of the cantilever and the thermal heating due to optical absorption are enhanced. This enhancement can be employed to control the deflection and the vibration amplitude of the cantilever. The thicknesses of the cantilever and the coating layer for maximal RP are shown to be 50–60 nm and 20–40 nm, respectively, when a laser with an optical wavelength 𝜆 > 500 nm is used. The RP amplification due to the cavity effect is most effective at 𝜆 ≃ 1200 nm and reaches a maximum value of 30. The heat absorption of the corresponding films is discussed. These results would be useful for optimizing the configurations of optical multilayer-based devices. © 2017 Elsevier B.V. All rights reserved.
1. Introduction: Optical fiber-based metrologies Optical fiber-based sensors are of intensive interest for use in performing measurements under critical conditions because of their fabrication simplicity in comparison to other available sensor devices [1–6]. These sensors, with their small heads, cause little interference with test samples and they can help to obtain high-accuracy results. Optical fibers can be combined with high-reflection surfaces to form a Fabry–Perot (FP) interferometer for displacement sensing [7–10] or they can be directly irradiated on the back of a cantilever to provide optical cavities for use in optomechanical laser cooling [11,12] (see Fig. 1(a)). Using this type of mechanism, Kim et al. [13] developed a cantilever-based optical interfacial force microscope to measure molecular interactions in liquids and obtained force resolution of less than 150 pN. Choi et al. [14] fabricated an FP sensor by cascading a photonic crystal fiber in combination with optical fibers for thermal sensing and temperatures of up to 1000 ◦ C could be measured using this optical cavity. Recently, Iannuzzi et al. [16,17] fabricated a fiber-top cantilever (FTC) by carving the end of an optical fiber [see Fig. 1(b)] and provided measurement abilities in both air and liquids that match those of
commercial atomic force microscopes [18–20]. A similar system was used by Li et al. [21] in which an FTC was implemented for temperature sensing at temperatures of more than 500◦ C. In these systems, the cantilever’s frequency shift and deflection are detected by observation of the interference between the light that is reflected at the cantilever and the light that is reflected at the coating layer with thickness 𝑡𝐶 at the fiber end, as per common optical fiber interferometers [9,22,23]. However, the roles of the coating thickness 𝑡𝐶 and the cantilever thickness 𝑡 have not been studied to date and the experimental basis for their fabrication is still open to question. The dimensions of the cantilever, for example, are determined based on its mechanical properties, which are planned to provide optimized readout data. Simultaneously, the use of a structure that can optimize the optical control of the cantilever has not been taken into account. In addition, the coating thickness shows an important role in the enhancement of the optical fields. Therefore, a full-scale study of the resonance effects, which are dependent on the coating, is necessary for accurate fabrication of optical cavities and FTCs. Multiple parallel thin films can be used as filters for optical devices with wavelength-dependent transmission [24,25] and their optical
* Corresponding address: Theoretical Physics Research Group and Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam.
E-mail address: [email protected] (N.D. Vy). http://dx.doi.org/10.1016/j.optcom.2017.07.023 Received 13 April 2017; Received in revised form 30 June 2017; Accepted 5 July 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 403 (2017) 150–154
Fig. 1. Optical fiber-based sensors. A fiber can be directly pointed on cantilever which is coated by a metallic layer to create an optical cavity [11,12,15] (a) or it can be carved at the end to create a fiber-top cantilever [16] (b). The light intensity is enhanced because photons can go back and forth several times via the reflection with metallic surfaces before they leak out the cavity. Model of a multilayer system that is irradiated by a laser beam (c). The incident and reflective field amplitudes of the 𝑗th layer are 𝐸𝑗,𝑖 and 𝐸𝑗,𝑟 . 𝐸𝑗,𝑟 = 0 for the last layer. Here, 𝑁 = 3 we have 𝑡𝐶 = 𝑑1 − 𝑑0 , 𝑡 = 𝑑3 − 𝑑2 , and 𝐸4,𝑟 = 0.
properties are generally examined using transfer matrices for each thin film, which allows total transmission/reflection to be obtained [25–27]. However, certain quantities for specific thin films, e.g., the field amplitudes inside the last film on the optical axis that are used to study the absorbed heat, or the field amplitudes immediately outside the film that are used to calculate the radiation pressure (RP), have not been evaluated to date. Therefore, the optimal configurations required for these multiple parallel thin films to optimize their mechanical and optical effects have not yet been determined. In this study, for the first time, we study the dependence of the optical properties of these optical multilayers on film thickness to allow optimal configurations to be achieved. It has been shown that a soft cantilever that is irradiated by a strongly localized force can have its vibration amplitude greatly changed [28], and that the modulated amplitude can be used to enhance the sensitivity of the cantilever when it is operated at higher order mechanical frequencies.
Fig. 2. Enhancement ratio 𝜂𝑟 [see Eq. (8)] for 𝜆 = 633 nm and 𝑡 = 80 nm for various coating thicknesses 𝑡𝐶 . Greatest enhancement is seen for 𝑡𝐶 ∼ 47 nm (white dashed line) and this is shown on the top panel (red solid line).
2. Analytical method th boundary, [ ] [ | 𝜖𝑗 𝐸𝑗,𝑖 𝑒𝑖𝑘𝑗 𝑧 + 𝐸𝑗,𝑟 𝑒−𝑖𝑘𝑗 𝑧 | = 𝜖𝑗+1 𝐸𝑗+1,𝑖 𝑒𝑖𝑘𝑗+1 𝑧 + |𝑧=𝑑𝑗 ] | + 𝐸𝑗+1,𝑟 𝑒−𝑖𝑘𝑗+1 𝑧 | , |𝑧=𝑑𝑗 [ ] [ d d | 𝜖𝑗 (𝐸 𝑒𝑖𝑘𝑗 𝑧 + 𝐸𝑗,𝑟 𝑒−𝑖𝑘𝑗 𝑧 )| = 𝜖𝑗+1 (𝐸 𝑒𝑖𝑘𝑗+1 𝑧 + |𝑧=𝑑𝑗 d𝑧 𝑗,𝑖 d𝑧 𝑗+1,𝑖 ] | + 𝐸𝑗+1,𝑟 𝑒−𝑖𝑘𝑗+1 𝑧 )| , |𝑧=𝑑𝑗
A system of several layers arranged in parallel to form an optical cavity is shown in Fig. 1(c). A laser which has a wavelength 𝜆 is assumed to propagate perpendicularly to the multilayer surfaces. The parallel component of the wave vector [29–32] is vanished, 𝑘∥ = 𝑘2𝑥 + 𝑘2𝑦 = 0, and only the perpendicular part remains, |𝐤| = 𝑘⟂ . The wavenumber is reduced to be 𝑘0 = 𝑘⟂ = 2𝜋∕𝜆 = 𝜔𝑜𝑝𝑡 ∕𝑐, where 𝜔𝑜𝑝𝑡 is the optical frequency. The material of the coating layer (𝑡𝐶 ) and of the cantilever layer (𝑡) are assumed to be gold which has a dielectric function 𝜖𝐴𝑢 . The cantilever back, which is made of silicon [Fig. 1(a)] or of glass [Fig. 1(b)], is assumed to have a unity dielectric function, 𝜖0 = 1. √ Inside the Au layers, the wavenumber is 𝑘𝐴𝑢 = 𝑠𝑘0 where 𝑠 = 𝜖𝐴𝑢 ∕𝜖0 and 𝜖𝐴𝑢[0] is the dielectric function of Au[air]. The complex dielectric function of Au is 𝜖𝐴𝑢 (𝜔𝑜𝑝𝑡 ) = 𝜖𝑏 − 𝜔2𝑝𝑙 ∕(𝜔2𝑜𝑝𝑡 + 𝑖𝛾𝜔𝑜𝑝𝑡 ), where 𝜔𝑜𝑝𝑡 = 2𝜋𝑐∕𝜆 is the laser frequency, 𝜖𝑏 is the background dielectric constant, 𝜔𝑝𝑙 is the bulk plasma frequency, 𝛾 is the nonradiative damping parameter, and 𝑐 is the speed of light. By writing out all required equations for the electromagnetic fields and applying the boundary conditions for the 𝑗
(1)
(2)
the field amplitudes 𝐸𝑗,𝑖 and 𝐸𝑗,𝑟 in every layer can then be obtained, where 𝑘𝑗 is the wavenumber in layer 𝑗 and 𝜖𝑗{𝑗+1} is the dielectric function. Using these thin films, the system can trap and enhance the magnitudes of the field intensities by several times [11]. The RP 𝑃𝑟𝑎𝑑 and the corresponding cavity-induced radiation force 𝐹𝑟𝑎𝑑 that are exerted on these thin films are usually calculated using the transfer matrix method. However, using of the multiplication of matrices gives lengthy expressions for the field amplitudes and it will be difficult to explicitly reveal the role of film thicknesses in field enhancement. Therefore, we 151
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Fig. 3. Enhancement ratio 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 180 nm and 𝜆 = 633 nm. Greatest enhancement is seen for 𝑡𝐶 = 40–60 nm. The 𝜂𝑟 of 𝑡 = 80 nm (black dashed line) and that of 𝑡 = 120 nm (red dash-dotted line) show a small difference (see top panel). A cut line at 𝑡𝐶 = 47 nm (black dash–dot-dotted line) is also presented on the right panel, which shows clearly the asymptotic behavior of 𝜂𝑟 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3. Field enhancement
explicitly reveal the roles of the coating thicknesses at the cleaved end (𝑡𝐶 ) and at the cantilever back (𝑡). The 𝐹𝑟𝑎𝑑 that is exerted on a film can then be calculated using 𝐹𝑟𝑎𝑑 =
∫𝑆
𝐓(𝐧)d𝑆,
First, the value of 𝜂𝑟 for 𝜆 = 633 nm (i.e., the wavelength of the He–Ne laser) and 𝑡 = 80 nm has been shown as an example in Fig. 2. The figure shows that there is a region around 𝑡𝐶 ≃ 47 nm where 𝜂𝑟 reaches a maximal value of ∼14 (indicated by the yellow dashed line) and this is clearly shown in the top panel (indicated by the red solid line). Increasing the value of 𝑡𝐶 reflects more of the light back to the fiber and thus reduces the total field intensity inside the cavity. Therefore, there is a set of optimal thicknesses for 𝑡 and 𝑡𝐶 that allows 𝜂𝑟 to be maximized. Second, a general view of 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 180 nm is shown in Fig. 3. The figure shows that the maximal 𝜂𝑟 range corresponds to 𝑡𝐶 = 40–60 nm. Specially, for values of 𝑡 ≥ 80 nm, 𝜂𝑟 begins to obtain asymptotic values and increasing 𝑡 (to 120 nm, for example, as indicated by the red dash-dotted line) does not enhance 𝜂𝑟 greatly in comparison to that at 𝑡 = 80 nm (indicated by the black dashed line); additionally, at 𝑡 = 80 nm, 𝜂𝑟 has a flat form for 𝑡𝐶 = 40–60 nm. Therefore, from an experimental viewpoint, we can freely choose a cantilever with a desired thickness, i.e., with desired mechanical properties, and with similar optical properties. If we choose the thickness at which 𝜂𝑟 reaches 90% of the asymptotic value of 𝜂𝑟 , i.e., 𝜂𝑟 in the case where 𝑡 → 𝜆, to be the optimal thickness, we can summarize the set of optimal parameters for Au at 633 as (𝜆, 𝑡, 𝑡𝐶 ) = (633, 80, 47) nm. Using of longer wavelengths shows greater enhancements of 𝜂𝑟 , as seen for 𝜆 = 1064 nm (i.e., the wavelength of the Nd:YAG laser) in Fig. 4. For the cantilever thickness of 𝑡 = 60 nm ∼ 𝜆∕16, 𝜂𝑟 reaches a large value of ∼28 and asymptotically reaches ∼30 for increasing value of 𝑡. The 𝜂𝑟 for 𝑡 = 80 nm (red dash-dotted line) increases slightly in comparison to that for 𝑡 = 60 nm (black dashed line) and has a nearly flat form for 𝑡𝐶 = 20–35 nm. Therefore, we can summarize the set of optimal parameters for Au at 1064 nm as (𝜆, 𝑡, 𝑡𝐶 ) = (1064, 60, 27) nm. Finally, we determine the full picture for the optimal coating thickness 𝑡𝐶 and the cantilever thickness 𝑡 for a wide range of optical wavelengths, where 𝜆 = 300–1500 nm [see Fig. 5]. The figure clearly shows that, for this metal (Au), the maximal enhancement ratio is ∼ 30 and this is achievable at long wavelengths, e.g., 𝜆 > 1000 nm. This can be explained based on the properties of the dielectric function at longer wavelengths, which are described above. It means for a given laser with
(3)
where T is the Maxwell stress tensor, 𝑆 is an arbitrary surface that surrounds the film [33], and n is the normal vector on 𝑆. The derived 𝐹𝑟𝑎𝑑 has been shown to be proportional to the difference between the field intensities on the left and right sides and can be expressed using the reflection and absorption coefficients [33,34]. For a film that is directly irradiated using a laser, 𝑃𝑟𝑎𝑑 = (2𝑅 + 𝐴)𝑃 ∕𝑐 = 𝜂𝑃 ∕𝑐, where 𝑅[𝐴] is the reflection[transmission] coefficient, 𝑃 is the input power, and 𝜂 can be called the impact coefficient, it measures the amount of momenta that a photon transfers to the reflective surface; e.g. 𝜂 = 2 for a total-reflection surface. For the simplest possible optical cavity, e.g., an FTC, 𝑁 = 3 [see Fig. 1(b) and (c)], and the impact coefficient is 𝜂𝑐𝑎𝑣 = (|𝐸2𝑖 |2 + |𝐸2𝑟 |2 − |𝐸𝑡 |2 )∕|𝐷|2 , where 𝐸2𝑖 = −4𝑒𝑖𝑘0 (𝑠−1)𝛼 𝑠[𝑒2𝑖𝑘𝐴𝑢 𝛽 (𝑠 − 1)2 − (𝑠 + 1)2 ]∕𝐷,
(4)
𝐸2𝑟 = 8𝑖𝑒𝑖𝑘0 (2𝛿+𝛼+𝑠𝛼+𝑠𝛽) sin(𝑘𝐴𝑢 𝛽)𝑠(𝑠2 − 1)∕𝐷,
(5)
𝐸4𝑖 = 16𝑒𝑖𝑘0 (𝑠−1)(𝛼+𝛽) 𝑠2 ∕𝐷,
(6)
and [ 𝐷 = (𝑠2 − 1)2 4𝑒𝑖𝑘0 (2𝛿+𝑠𝛼+𝑠𝛽) sin(𝑘𝐴𝑢 𝛼) sin(𝑘𝐴𝑢 𝛽) − 𝑒2𝑖𝑘𝐴𝑢 𝛼 + ] − 𝑒2𝑖𝑘𝐴𝑢 𝛽 + 𝑒2𝑖𝑘𝐴𝑢 (𝛼+𝛽) (𝑠 − 1)4 + (𝑠 + 1)4 .
(7)
In the equations above, we have normalized the dimensions of the coating layer and the cantilever with respect to the cavity length 𝐿𝑐 , as follows: 𝛼 = 𝑡𝑐 ∕𝐿𝑐 for the coating thickness, 𝛽 = 𝑡∕𝐿𝑐 for the cantilever thickness, and 𝛿 = 𝑑∕𝐿𝑐 for detuning from the resonance position. Additionally, we define the enhancement ratio 𝜂𝑟 = 𝜂𝑐𝑎𝑣 ∕𝜂.
(8)
as the amplification factor that the cavity can create on the cantilever. This amplification process is a resonance effect that occurs in optical multilayers in which the effective cavity length is a multiple of half the operating wavelength. 152
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Fig. 4. Enhancement ratio 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 130 nm and 𝜆 = 1064 nm. Greatest enhancement is seen for 𝑡𝐶 = 20–35 nm. The 𝜂𝑟 of 𝑡 = 80 nm (red dash-dotted line) and that of 𝑡 = 60 nm (black dashed line) show a small difference (see top panel). A cut line at 𝑡𝐶 = 27 nm (black dash–dot-dotted line) is also presented on the right panel, which shows clearly the asymptotic behavior of 𝜂𝑟 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
as shown in Ref. [35], where the integration is taken over t. The maxima of 𝑃𝑎𝑏𝑠 have been shown to be proportional to that of 𝐹𝑟𝑎𝑑 and have Lorentzian forms whose peaks position are coinciding with that of 𝐹𝑟𝑎𝑑 . This means that the optimal configuration for 𝐹𝑟𝑎𝑑 is also the optimal one for 𝑃𝑎𝑏𝑠 . Therefore, the above configuration can be the optimal one for other intensity-dependent optical effects. For the Au film thickness larger than 20 nm that is used in this study, the parameters in the dielectric function 𝜖𝐴𝑢 of the Au film have been assumed to be constants. For ultra-thin films which have thicknesses less than 20 nm, a decrease of plasma frequency 𝜔𝑝𝑙 for decreasing of film thickness [36] (𝑡, 𝑡𝐶 < 20 nm) leads to a decrease of permittivity, and this size effects should be taken into account. This reduces the imaginary part, Im[𝜖𝐴𝑢 ] = 𝜔2𝑝𝑙 𝛾𝜔𝑜𝑝𝑡 ∕(𝜔4𝑜𝑝𝑡 + 𝛾 2 𝜔2𝑜𝑝𝑡 ), and increase the real part, 𝜖𝑏 −𝜔2𝑝𝑙 𝜔2 ∕(𝜔4𝑜𝑝𝑡 +𝛾 2 𝜔2𝑜𝑝𝑡 ), of 𝜖𝐴𝑢 as shown by Wang et al. [37]. This effect is due to the discontinuous structure of ultra-thin films, i.e., the film is no longer a continuous medium but it has the topology of heterogeneous clusters that has strong plasmonic effects. Therefore, for ultra-thin films, peaks of the plasma resonance in Fig. 5 will shift to the long-wavelength region. Especially, an increase of the imaginary part simultaneously with a decrease of the real part of the dielectric function 𝜖𝐴𝑢 implies an enhancement of the optical absorption on thin films. And the above Maxwell’s equations cannot be used accurately, i.e., when this plasmonic effect is strong. Here, we use the parameter for film thicknesses which are greater than 20 nm, 𝜖𝑏 = 12, 𝜔𝑝𝑙 = 8.95 eV, and 𝛾 = 72.3 meV. Cantilever length is 233 μm and width is 23 μm. The heat absorption of the back layer (a part of the optical fiber) of the cantilever is assumed to be negligible.
Fig. 5. (Left axis) Thicknesses of the cantilever metallic film 𝑡 (black solid line) and of the coating layer 𝑡𝐶 (red dashed line) to maximize the enhancement ratio 𝜂𝑟 (blue dots, right axis). The dip at 𝜆 ∼ 450 nm corresponds to the absorption band of Au where the field intensity difference between the left and right sides is small. The figure shows that increasing values of 𝑡 > 140 nm or 𝑡𝐶 > 60 nm does not increase 𝜂𝑟 .
wavelength 𝜆, the film thicknesses should be chosen around the range of 𝑡 for the cantilever and 𝑡𝐶 for the coating for a maximal enhancement ratio of RP. In general, the metallic layer on the cantilever (𝑡, black solid line) is ∼30 nm thicker than the fiber-end coating layer (𝑡𝐶 , red dashed line). For example, if the input laser has 𝜆 = 1200 nm, 𝑡 should be ∼55 nm and 𝑡𝐶 ∼ 25 nm, and 𝜂𝑟 reaches 30, a great enhancement of RP. In particular, at the plasmon resonance frequency, where Re[𝜖(𝜔𝑜𝑝𝑡 ) ] = 0, 𝜔2𝑟𝑒𝑠 = 𝜔2𝑝𝑙 ∕𝜖𝑏 − 𝛾 2 , and this leads to 𝜆𝑟𝑒𝑠 = ℎ𝑐∕𝜔𝑟𝑒𝑠 ≃ 450 nm (indicated by the green dotted line), and Maxwell’s equations cannot be solved exactly using the conventional method above; therefore, 𝜂𝑟 cannot be obtained because the optical energy is now transferred to the plasmon oscillations of the electrons. However, approximate values can be deduced, and in Fig. 5 we plot a full range of 𝜂𝑟 that involves 𝜆 < 𝜆𝑟𝑒𝑠 and 𝜆 ≃ 𝜆𝑟𝑒𝑠 to explicitly demonstrate such a resonance effect. The absorbed heat on the cantilever can be given by 𝑑3 =𝑑2 +𝑡
𝑃𝑎𝑏𝑠 =
∫𝑑2
𝑑𝑧|𝐸3 |2 𝐈𝐦[1 − 𝜖𝐴𝑢 ]𝜔𝑜𝑝𝑡 ∕𝑐,
4. Conclusion In summary, we have determined optimal parameters for a system of planar optical multilayers to allow both the radiation pressure and the thermal heating of these cavities to be enhanced greatly. The optimal coating and cantilever thicknesses for two typical input laser wavelengths of 633 nm and 1064 nm have been deduced. A figure for the enhancement factor of the radiation pressure in a wide range of optical wavelength, 300–1500 nm, has been shown. As a result, we obtain a set of parameters for gold thin films in measurement systems such as the fiber-top cantilever. While gold has been used as a typical material to examine the trapping effects, the results that are reported here are universal and are applicable to similar metals such as Ag.
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Acknowledgments
[18] D. Iannuzzi, S. Deladi, J.W. Berenschot, S. de Man, K. Heeck, M.C. Elwenspoek, Fibertop atomic force microscope, Rev. Sci. Instrum. 77 (10) (2006) 106105. http://dx. doi.org/10.1063/1.2358710. URL http://scitation.aip.org/content/aip/journal/rsi/ N.D. Vy is thankful to Associate Professor Ho Trung D. and Professor 77/10/10.1063/1.2358710. L.V. Hoang for their encouragement. This research is funded by Vietnam [19] K.B. Gavan, J.H. Rector, K. Heeck, D. Chavan, G. Gruca, T.H. Oosterkamp, National Foundation for Science and Technology Development (NAFOSD. Iannuzzi, Top-down approach to fiber-top cantilevers, Opt. Lett. 36 (15) (2011) 2898–2900. http://dx.doi.org/10.1364/OL.36.002898. URL http://ol.osa. TED) under grant number 103.01-2016.70. org/abstract.cfm?URI=ol-36-15-2898. [20] M. Cui, C.H. Van Hoorn, D. Ianuuzzi, Miniaturized fibre-top cantilevers on etched References fibres, J. Micros. 264 (3) (2016) 370–374. http://dx.doi.org/10.1111/jmi.12452. [21] J. Li, F. Albri, J.N. Sun, M.M. Miliar, R.R.J. Maier, D.P. Hand, W.N. MacPherson, [1] L. Liu, S. Peng, W. Wen, P. Sheng, Micro thermoindicators and optical-electronic Fabricating optical fibre-top cantilevers for temperature sensing, Meas. Sci. Tech. temperature control for microfluidic applications, Appl. Phys. Lett. 91 (9) (2007) 25 (3) (2014) 035206. URL http://stacks.iop.org/0957-0233/25/i=3/a=035206. 093513. http://dx.doi.org/10.1063/1.2776848. [22] D. Rugar, H.J. Mamin, R. Erlandsson, J.E. Stern, B.D. Terris, Force microscope using [2] C. Wu, H.Y. Fu, K.K. Qureshi, B.-O. Guan, H.Y. Tam, High-pressure and higha fiberoptic displacement sensor, Rev. Sci. Instrum. 59 (11) (1988) 2337–2340. temperature characteristics of a Fabry–Perot interferometer based on photonic http://dx.doi.org/10.1063/1.1139958. crystal fiber, Opt. Lett. 36 (3) (2011) 412–414. http://dx.doi.org/10.1364/OL.36. [23] D. Rugar, H.J. Mamin, P. Guethner, Improved fiberoptic interferometer for atomic 000412. URL http://ol.osa.org/abstract.cfm?URI=ol-36-3-412. force microscopy, Appl. Phys. Lett. 55 (25) (1989) 2588–2590. http://dx.doi.org/ [3] X. Zhang, W. Peng, Temperature-independent fiber salinity sensor based on Fabry10.1063/1.101987. Perot interference, Opt. Express 23 (8) (2015) 10353–10358. http://dx.doi.org/ [24] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, 10.1364/OE.23.010353. URL http://www.opticsexpress.org/abstract.cfm?URI=oeInterference and Diffraction of Light, Cambridge University Press, Cambridge, UK, 23-8-10353. 1997. URL https://books.google.com.vn/books/about/Principles_of_Optics.html?id [4] N.H. Wan, F. Meng, T. Schröder, R.-J. Shiue, E.H. Chen, D. Englund, High-resolution =oV80AAAAIAAJ&redir_esc=y. optical spectroscopy using multimode interference in a compact tapered fibre, [25] H. MacLeod, Thin-Film Optical Filters, third ed., in: Series in Optics and OptoelecNature Commun. 6 (2015) 7762. http://dx.doi.org/10.1038/ncomms8762. tronics, CRC Press, 2001. URL https://books.google.com.vn/books?id=D0S9hxzPJq8C. [5] J. Li, F. Albri, R.R.J. Maier, W. Shu, J. Sun, D.P. Hand, W.N. MacPherson, A micro[26] N. Duy Vy, T. Iida, Ambient-dependent optomechanical control of cantilever with machined optical fiber cantilever as a miniaturized ph sensor, IEEE Sens. J. 15 (12) mechanical nonlinearity by cavity-induced radiation force, Appl. Phys. Lett. 102 (9) (2015) 7221–7228. http://dx.doi.org/10.1109/JSEN.2015.2472971. (2013) 091101. URL http://scitation.aip.org/content/aip/journal/apl/102/9/10. [6] Y. Liu, D.N. Wang, W.P. Chen, Crescent shaped Fabry-Perot fiber cavity for ultra1063/1.4794060. sensitive strain measurement, Sci. Rep. 6 (2016) 38390. http://dx.doi.org/10.1038/ [27] N.D. Vy, L. Tri Dat, T. Iida, Cancelation of thermally induced frequency shifts in srep38390. URL http://www.nature.com/articles/srep38390. bimaterial cantilevers by nonlinear optomechanical interactions, Appl. Phys. Lett. [7] A. Kersey, D. Jackson, M. Corke, A simple fibre Fabry-Perot sensor, Opt. Commun. 109 (5) (2016). http://dx.doi.org/10.1063/1.4960380. URL http://scitation.aip. 45 (2) (1983) 71–74. http://dx.doi.org/10.1016/0030-4018(83)90047-0. URL http: org/content/aip/journal/apl/109/5/10.1063/1.4960380. //www.sciencedirect.com/science/article/pii/0030401883900470. [28] C.M. Hoang, N.D. Vy, L.T. Dat, T. Iida, Enhancing amplitudes of higher-order [8] J.L. Santos, A.P. Leite, D.A. Jackson, Optical fiber sensing with a low-finesse Fabry– eigenmodes of atomic force microscope cantilevers by laser for better mass sensing, Perot cavity, Appl. Opt. 31 (34) (1992) 7361–7366. http://dx.doi.org/10.1364/AO. Japan. J. Appl. Phys. 56 (6S1) (2017) 06GK05. URL http://stacks.iop.org/134731.007361. URL http://ao.osa.org/abstract.cfm?URI=ao-31-34-7361. 4065/56/i=6S1/a=06GK05. [9] T. Wang, S. Zheng, Z. Yang, A high precision displacement sensor using a low[29] M. Bordag, U. Mohideen, V. Mostepanenko, New developments in the Casimir effect, finesse fiber-optic Fabry-Pérot interferometer, Sens. Actuators A 69 (2) (1998) Phys. Rep. 353 (13) (2001) 1–205. http://dx.doi.org/10.1016/S0370-1573(01) 134–138. http://dx.doi.org/10.1016/S0924-4247(98)00075-2. URL http://www. 00015-1. URL http://www.sciencedirect.com/science/article/pii/S0370157301000 sciencedirect.com/science/article/pii/S0924424798000752. 151. [10] L. Perret, L. Chassagne, S. Topu¸, P. Ruaux, B. Cagneau, Y. Alayli, Fiber optics sensor [30] C. Genet, A. Lambrecht, P.M. Neto, S. Reynaud, The casimir force between rough for sub-nanometric displacement and wide bandwidth systems, Sens. Actuators A metallic plates, Europhys. Lett. 62 (4) (2003) 484. http://stacks.iop.org/0295-5075/ 165 (2) (2011) 189–193. http://dx.doi.org/10.1016/j.sna.2010.10.010. URL http: 62/i=4/a=484. //www.sciencedirect.com/science/article/pii/S0924424710004565. [31] V.A. Parsegian, Van der Waals Forces, Cambridge University Press, 2005. URL [11] C.H. Metzger, K. Karrai, Cavity cooling of a microlever, Nature 432 (2004) 1002. https://books.google.com.vn/books?id=K4F7uHJVEOoC. http://dx.doi.org/10.1038/nature03118. URL https://www.nature.com/nature/journal/ [32] M. Bordag, I.V. Fialkovsky, D.M. Gitman, D.V. Vassilevich, Casimir interaction v432/n7020/full/nature03118.html. between a perfect conductor and graphene described by the dirac model, Phys. [12] F. Marquardt, S.M. Girvin, Optomechanics, Physics 2 (2009) 40. http://dx.doi.org/ Rev. B 80 (2009) 245406. http://dx.doi.org/10.1103/PhysRevB.80.245406. URL 10.1103/Physics.2.40. URL http://link.aps.org/doi/10.1103/Physics.2.40. https://link.aps.org/doi/10.1103/PhysRevB.80.245406. [13] B.I. Kim, L. Smith, T. Tran, S. Rossland, E. Parkinson, Cantilever-based optical [33] T. Iida, H. Ishihara, Study of the mechanical interaction between an electromagnetic interfacial force microscope in liquid using an optical-fiber tip, AIP Adv. 3 (3) (2013) field and a nanoscopic thin film near electronic resonance, Opt. Lett. 27 (2002) 754. 032126. http://dx.doi.org/10.1063/1.4795761. http://dx.doi.org/10.1364/OL.27.000754. [14] H.Y. Choi, K.S. Park, S.J. Park, U.-C. Paek, B.H. Lee, E.S. Choi, Miniature fiber-optic [34] A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: high temperature sensor based on a hybrid structured Fabry–Perot interferometer, A Reprint Volume with Commentaries, 2006. URL https://books.google.com.vn/ Opt. Lett. 33 (21) (2008) 2455–2457. http://dx.doi.org/10.1364/OL.33.002455. books?id=VGzVCgAAQBAJ. URL http://ol.osa.org/abstract.cfm?URI=ol-33-21-2455. [35] N.D. Vy, T. Iida, Enhancing thermally induced effects on atomic force microscope [15] N.D. Vy, T. Iida, Nonlinear analysis of sub-millikelvin optomechanical cooling for cantilevers using optical microcavities, Appl. Phys. Express 9 (12) (2016) 126601. extremely low noise quantum measurement, Appl. Phys. Express 8 (3) (2015) URL http://stacks.iop.org/1882-0786/9/i=12/a=126601. 032801. URL http://stacks.iop.org/1882-0786/8/i=3/a=032801. [36] S. Laref, J. Cao, A. Asaduzzaman, K. Runge, P. Deymier, R.W. Ziolkowski, [16] D. Iannuzzi, S. Deladi, V.J. Gadgil, R.G.P. Sanders, H. Schreuders, M.C. Elwenspoek, M. Miyawaki, K. Muralidharan, Size-dependent permittivity and intrinsic optical Monolithic fiber-top sensor for critical environments and standard applications, anisotropy of nanometric gold thin films: a density functional theory study, Opt. Appl. Phys. Lett. 88 (5) (2006) 053501. http://dx.doi.org/10.1063/1.2170139. URL Express 21 (10) (2013) 11827–11838. http://dx.doi.org/10.1364/OE.21.011827. http://scitation.aip.org/content/aip/journal/apl/88/5/10.1063/1.2170139. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-21-10-11827. [17] S. Deladi, D. Iannuzzi, V.J. Gadgil, H. Schreuders, M.C. Elwenspoek, Carving fiber[37] X. Wang, K. ping Chen, M. Zhao, D.D. Nolte, Refractive index and dielectric top optomechanical transducers from an optical fiber, J. Micromech. Microeng. constant evolution of ultra-thin gold from clusters to films, Opt. Express 18 (24) 16 (5) (2006) 886. URL http://stacks.iop.org/0960-1317/16/i=5/a=002. (2010) 24859–24867. http://dx.doi.org/10.1364/OE.18.024859. URL http://www. opticsexpress.org/abstract.cfm?URI=oe-18-24-24859.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Optimal coating thickness for enhancement of optical effects in optical multilayer-based metrologies Chu Manh Hoang a , Takuya Iida b,c , Le Tri Dat d , Ho Thanh Huy d , Nguyen Duy Vy e,f, * a
International Training Institute for Materials Science, Hanoi University of Science and Technology, No. 1, Dai Co Viet, Hai Ba Trung, Hanoi, Viet Nam Department of Physical Science, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan c Research Institute for Light-induced Acceleration System (RILACS), Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan d Faculty of Physics and Engineering Physics, University of Science, Ho Chi Minh City, Viet Nam e Theoretical Physics Research Group, Ton Duc Thang University, Ho Chi Minh City, Viet Nam f Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b
a r t i c l e
i n f o
Keywords: Fiber-top cantilever Optical multilayer Atomic force microscope Radiation pressure
a b s t r a c t We theoretically determine an optimal configuration for a planar optical multilayer that is used to enhance the radiation pressure (RP) and the thermal heating in fiber-based metrology systems such as fiber-top cantilevers. The RP exerting on the metallic layer of the cantilever and the thermal heating due to optical absorption are enhanced. This enhancement can be employed to control the deflection and the vibration amplitude of the cantilever. The thicknesses of the cantilever and the coating layer for maximal RP are shown to be 50–60 nm and 20–40 nm, respectively, when a laser with an optical wavelength 𝜆 > 500 nm is used. The RP amplification due to the cavity effect is most effective at 𝜆 ≃ 1200 nm and reaches a maximum value of 30. The heat absorption of the corresponding films is discussed. These results would be useful for optimizing the configurations of optical multilayer-based devices. © 2017 Elsevier B.V. All rights reserved.
1. Introduction: Optical fiber-based metrologies Optical fiber-based sensors are of intensive interest for use in performing measurements under critical conditions because of their fabrication simplicity in comparison to other available sensor devices [1–6]. These sensors, with their small heads, cause little interference with test samples and they can help to obtain high-accuracy results. Optical fibers can be combined with high-reflection surfaces to form a Fabry–Perot (FP) interferometer for displacement sensing [7–10] or they can be directly irradiated on the back of a cantilever to provide optical cavities for use in optomechanical laser cooling [11,12] (see Fig. 1(a)). Using this type of mechanism, Kim et al. [13] developed a cantilever-based optical interfacial force microscope to measure molecular interactions in liquids and obtained force resolution of less than 150 pN. Choi et al. [14] fabricated an FP sensor by cascading a photonic crystal fiber in combination with optical fibers for thermal sensing and temperatures of up to 1000 ◦ C could be measured using this optical cavity. Recently, Iannuzzi et al. [16,17] fabricated a fiber-top cantilever (FTC) by carving the end of an optical fiber [see Fig. 1(b)] and provided measurement abilities in both air and liquids that match those of
commercial atomic force microscopes [18–20]. A similar system was used by Li et al. [21] in which an FTC was implemented for temperature sensing at temperatures of more than 500◦ C. In these systems, the cantilever’s frequency shift and deflection are detected by observation of the interference between the light that is reflected at the cantilever and the light that is reflected at the coating layer with thickness 𝑡𝐶 at the fiber end, as per common optical fiber interferometers [9,22,23]. However, the roles of the coating thickness 𝑡𝐶 and the cantilever thickness 𝑡 have not been studied to date and the experimental basis for their fabrication is still open to question. The dimensions of the cantilever, for example, are determined based on its mechanical properties, which are planned to provide optimized readout data. Simultaneously, the use of a structure that can optimize the optical control of the cantilever has not been taken into account. In addition, the coating thickness shows an important role in the enhancement of the optical fields. Therefore, a full-scale study of the resonance effects, which are dependent on the coating, is necessary for accurate fabrication of optical cavities and FTCs. Multiple parallel thin films can be used as filters for optical devices with wavelength-dependent transmission [24,25] and their optical
* Corresponding address: Theoretical Physics Research Group and Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam.
E-mail address: [email protected] (N.D. Vy). http://dx.doi.org/10.1016/j.optcom.2017.07.023 Received 13 April 2017; Received in revised form 30 June 2017; Accepted 5 July 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Fig. 1. Optical fiber-based sensors. A fiber can be directly pointed on cantilever which is coated by a metallic layer to create an optical cavity [11,12,15] (a) or it can be carved at the end to create a fiber-top cantilever [16] (b). The light intensity is enhanced because photons can go back and forth several times via the reflection with metallic surfaces before they leak out the cavity. Model of a multilayer system that is irradiated by a laser beam (c). The incident and reflective field amplitudes of the 𝑗th layer are 𝐸𝑗,𝑖 and 𝐸𝑗,𝑟 . 𝐸𝑗,𝑟 = 0 for the last layer. Here, 𝑁 = 3 we have 𝑡𝐶 = 𝑑1 − 𝑑0 , 𝑡 = 𝑑3 − 𝑑2 , and 𝐸4,𝑟 = 0.
properties are generally examined using transfer matrices for each thin film, which allows total transmission/reflection to be obtained [25–27]. However, certain quantities for specific thin films, e.g., the field amplitudes inside the last film on the optical axis that are used to study the absorbed heat, or the field amplitudes immediately outside the film that are used to calculate the radiation pressure (RP), have not been evaluated to date. Therefore, the optimal configurations required for these multiple parallel thin films to optimize their mechanical and optical effects have not yet been determined. In this study, for the first time, we study the dependence of the optical properties of these optical multilayers on film thickness to allow optimal configurations to be achieved. It has been shown that a soft cantilever that is irradiated by a strongly localized force can have its vibration amplitude greatly changed [28], and that the modulated amplitude can be used to enhance the sensitivity of the cantilever when it is operated at higher order mechanical frequencies.
Fig. 2. Enhancement ratio 𝜂𝑟 [see Eq. (8)] for 𝜆 = 633 nm and 𝑡 = 80 nm for various coating thicknesses 𝑡𝐶 . Greatest enhancement is seen for 𝑡𝐶 ∼ 47 nm (white dashed line) and this is shown on the top panel (red solid line).
2. Analytical method th boundary, [ ] [ | 𝜖𝑗 𝐸𝑗,𝑖 𝑒𝑖𝑘𝑗 𝑧 + 𝐸𝑗,𝑟 𝑒−𝑖𝑘𝑗 𝑧 | = 𝜖𝑗+1 𝐸𝑗+1,𝑖 𝑒𝑖𝑘𝑗+1 𝑧 + |𝑧=𝑑𝑗 ] | + 𝐸𝑗+1,𝑟 𝑒−𝑖𝑘𝑗+1 𝑧 | , |𝑧=𝑑𝑗 [ ] [ d d | 𝜖𝑗 (𝐸 𝑒𝑖𝑘𝑗 𝑧 + 𝐸𝑗,𝑟 𝑒−𝑖𝑘𝑗 𝑧 )| = 𝜖𝑗+1 (𝐸 𝑒𝑖𝑘𝑗+1 𝑧 + |𝑧=𝑑𝑗 d𝑧 𝑗,𝑖 d𝑧 𝑗+1,𝑖 ] | + 𝐸𝑗+1,𝑟 𝑒−𝑖𝑘𝑗+1 𝑧 )| , |𝑧=𝑑𝑗
A system of several layers arranged in parallel to form an optical cavity is shown in Fig. 1(c). A laser which has a wavelength 𝜆 is assumed to propagate perpendicularly to the multilayer surfaces. The parallel component of the wave vector [29–32] is vanished, 𝑘∥ = 𝑘2𝑥 + 𝑘2𝑦 = 0, and only the perpendicular part remains, |𝐤| = 𝑘⟂ . The wavenumber is reduced to be 𝑘0 = 𝑘⟂ = 2𝜋∕𝜆 = 𝜔𝑜𝑝𝑡 ∕𝑐, where 𝜔𝑜𝑝𝑡 is the optical frequency. The material of the coating layer (𝑡𝐶 ) and of the cantilever layer (𝑡) are assumed to be gold which has a dielectric function 𝜖𝐴𝑢 . The cantilever back, which is made of silicon [Fig. 1(a)] or of glass [Fig. 1(b)], is assumed to have a unity dielectric function, 𝜖0 = 1. √ Inside the Au layers, the wavenumber is 𝑘𝐴𝑢 = 𝑠𝑘0 where 𝑠 = 𝜖𝐴𝑢 ∕𝜖0 and 𝜖𝐴𝑢[0] is the dielectric function of Au[air]. The complex dielectric function of Au is 𝜖𝐴𝑢 (𝜔𝑜𝑝𝑡 ) = 𝜖𝑏 − 𝜔2𝑝𝑙 ∕(𝜔2𝑜𝑝𝑡 + 𝑖𝛾𝜔𝑜𝑝𝑡 ), where 𝜔𝑜𝑝𝑡 = 2𝜋𝑐∕𝜆 is the laser frequency, 𝜖𝑏 is the background dielectric constant, 𝜔𝑝𝑙 is the bulk plasma frequency, 𝛾 is the nonradiative damping parameter, and 𝑐 is the speed of light. By writing out all required equations for the electromagnetic fields and applying the boundary conditions for the 𝑗
(1)
(2)
the field amplitudes 𝐸𝑗,𝑖 and 𝐸𝑗,𝑟 in every layer can then be obtained, where 𝑘𝑗 is the wavenumber in layer 𝑗 and 𝜖𝑗{𝑗+1} is the dielectric function. Using these thin films, the system can trap and enhance the magnitudes of the field intensities by several times [11]. The RP 𝑃𝑟𝑎𝑑 and the corresponding cavity-induced radiation force 𝐹𝑟𝑎𝑑 that are exerted on these thin films are usually calculated using the transfer matrix method. However, using of the multiplication of matrices gives lengthy expressions for the field amplitudes and it will be difficult to explicitly reveal the role of film thicknesses in field enhancement. Therefore, we 151
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Fig. 3. Enhancement ratio 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 180 nm and 𝜆 = 633 nm. Greatest enhancement is seen for 𝑡𝐶 = 40–60 nm. The 𝜂𝑟 of 𝑡 = 80 nm (black dashed line) and that of 𝑡 = 120 nm (red dash-dotted line) show a small difference (see top panel). A cut line at 𝑡𝐶 = 47 nm (black dash–dot-dotted line) is also presented on the right panel, which shows clearly the asymptotic behavior of 𝜂𝑟 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3. Field enhancement
explicitly reveal the roles of the coating thicknesses at the cleaved end (𝑡𝐶 ) and at the cantilever back (𝑡). The 𝐹𝑟𝑎𝑑 that is exerted on a film can then be calculated using 𝐹𝑟𝑎𝑑 =
∫𝑆
𝐓(𝐧)d𝑆,
First, the value of 𝜂𝑟 for 𝜆 = 633 nm (i.e., the wavelength of the He–Ne laser) and 𝑡 = 80 nm has been shown as an example in Fig. 2. The figure shows that there is a region around 𝑡𝐶 ≃ 47 nm where 𝜂𝑟 reaches a maximal value of ∼14 (indicated by the yellow dashed line) and this is clearly shown in the top panel (indicated by the red solid line). Increasing the value of 𝑡𝐶 reflects more of the light back to the fiber and thus reduces the total field intensity inside the cavity. Therefore, there is a set of optimal thicknesses for 𝑡 and 𝑡𝐶 that allows 𝜂𝑟 to be maximized. Second, a general view of 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 180 nm is shown in Fig. 3. The figure shows that the maximal 𝜂𝑟 range corresponds to 𝑡𝐶 = 40–60 nm. Specially, for values of 𝑡 ≥ 80 nm, 𝜂𝑟 begins to obtain asymptotic values and increasing 𝑡 (to 120 nm, for example, as indicated by the red dash-dotted line) does not enhance 𝜂𝑟 greatly in comparison to that at 𝑡 = 80 nm (indicated by the black dashed line); additionally, at 𝑡 = 80 nm, 𝜂𝑟 has a flat form for 𝑡𝐶 = 40–60 nm. Therefore, from an experimental viewpoint, we can freely choose a cantilever with a desired thickness, i.e., with desired mechanical properties, and with similar optical properties. If we choose the thickness at which 𝜂𝑟 reaches 90% of the asymptotic value of 𝜂𝑟 , i.e., 𝜂𝑟 in the case where 𝑡 → 𝜆, to be the optimal thickness, we can summarize the set of optimal parameters for Au at 633 as (𝜆, 𝑡, 𝑡𝐶 ) = (633, 80, 47) nm. Using of longer wavelengths shows greater enhancements of 𝜂𝑟 , as seen for 𝜆 = 1064 nm (i.e., the wavelength of the Nd:YAG laser) in Fig. 4. For the cantilever thickness of 𝑡 = 60 nm ∼ 𝜆∕16, 𝜂𝑟 reaches a large value of ∼28 and asymptotically reaches ∼30 for increasing value of 𝑡. The 𝜂𝑟 for 𝑡 = 80 nm (red dash-dotted line) increases slightly in comparison to that for 𝑡 = 60 nm (black dashed line) and has a nearly flat form for 𝑡𝐶 = 20–35 nm. Therefore, we can summarize the set of optimal parameters for Au at 1064 nm as (𝜆, 𝑡, 𝑡𝐶 ) = (1064, 60, 27) nm. Finally, we determine the full picture for the optimal coating thickness 𝑡𝐶 and the cantilever thickness 𝑡 for a wide range of optical wavelengths, where 𝜆 = 300–1500 nm [see Fig. 5]. The figure clearly shows that, for this metal (Au), the maximal enhancement ratio is ∼ 30 and this is achievable at long wavelengths, e.g., 𝜆 > 1000 nm. This can be explained based on the properties of the dielectric function at longer wavelengths, which are described above. It means for a given laser with
(3)
where T is the Maxwell stress tensor, 𝑆 is an arbitrary surface that surrounds the film [33], and n is the normal vector on 𝑆. The derived 𝐹𝑟𝑎𝑑 has been shown to be proportional to the difference between the field intensities on the left and right sides and can be expressed using the reflection and absorption coefficients [33,34]. For a film that is directly irradiated using a laser, 𝑃𝑟𝑎𝑑 = (2𝑅 + 𝐴)𝑃 ∕𝑐 = 𝜂𝑃 ∕𝑐, where 𝑅[𝐴] is the reflection[transmission] coefficient, 𝑃 is the input power, and 𝜂 can be called the impact coefficient, it measures the amount of momenta that a photon transfers to the reflective surface; e.g. 𝜂 = 2 for a total-reflection surface. For the simplest possible optical cavity, e.g., an FTC, 𝑁 = 3 [see Fig. 1(b) and (c)], and the impact coefficient is 𝜂𝑐𝑎𝑣 = (|𝐸2𝑖 |2 + |𝐸2𝑟 |2 − |𝐸𝑡 |2 )∕|𝐷|2 , where 𝐸2𝑖 = −4𝑒𝑖𝑘0 (𝑠−1)𝛼 𝑠[𝑒2𝑖𝑘𝐴𝑢 𝛽 (𝑠 − 1)2 − (𝑠 + 1)2 ]∕𝐷,
(4)
𝐸2𝑟 = 8𝑖𝑒𝑖𝑘0 (2𝛿+𝛼+𝑠𝛼+𝑠𝛽) sin(𝑘𝐴𝑢 𝛽)𝑠(𝑠2 − 1)∕𝐷,
(5)
𝐸4𝑖 = 16𝑒𝑖𝑘0 (𝑠−1)(𝛼+𝛽) 𝑠2 ∕𝐷,
(6)
and [ 𝐷 = (𝑠2 − 1)2 4𝑒𝑖𝑘0 (2𝛿+𝑠𝛼+𝑠𝛽) sin(𝑘𝐴𝑢 𝛼) sin(𝑘𝐴𝑢 𝛽) − 𝑒2𝑖𝑘𝐴𝑢 𝛼 + ] − 𝑒2𝑖𝑘𝐴𝑢 𝛽 + 𝑒2𝑖𝑘𝐴𝑢 (𝛼+𝛽) (𝑠 − 1)4 + (𝑠 + 1)4 .
(7)
In the equations above, we have normalized the dimensions of the coating layer and the cantilever with respect to the cavity length 𝐿𝑐 , as follows: 𝛼 = 𝑡𝑐 ∕𝐿𝑐 for the coating thickness, 𝛽 = 𝑡∕𝐿𝑐 for the cantilever thickness, and 𝛿 = 𝑑∕𝐿𝑐 for detuning from the resonance position. Additionally, we define the enhancement ratio 𝜂𝑟 = 𝜂𝑐𝑎𝑣 ∕𝜂.
(8)
as the amplification factor that the cavity can create on the cantilever. This amplification process is a resonance effect that occurs in optical multilayers in which the effective cavity length is a multiple of half the operating wavelength. 152
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Fig. 4. Enhancement ratio 𝜂𝑟 for various film thicknesses in the range 𝑡 ≤ 130 nm and 𝜆 = 1064 nm. Greatest enhancement is seen for 𝑡𝐶 = 20–35 nm. The 𝜂𝑟 of 𝑡 = 80 nm (red dash-dotted line) and that of 𝑡 = 60 nm (black dashed line) show a small difference (see top panel). A cut line at 𝑡𝐶 = 27 nm (black dash–dot-dotted line) is also presented on the right panel, which shows clearly the asymptotic behavior of 𝜂𝑟 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
as shown in Ref. [35], where the integration is taken over t. The maxima of 𝑃𝑎𝑏𝑠 have been shown to be proportional to that of 𝐹𝑟𝑎𝑑 and have Lorentzian forms whose peaks position are coinciding with that of 𝐹𝑟𝑎𝑑 . This means that the optimal configuration for 𝐹𝑟𝑎𝑑 is also the optimal one for 𝑃𝑎𝑏𝑠 . Therefore, the above configuration can be the optimal one for other intensity-dependent optical effects. For the Au film thickness larger than 20 nm that is used in this study, the parameters in the dielectric function 𝜖𝐴𝑢 of the Au film have been assumed to be constants. For ultra-thin films which have thicknesses less than 20 nm, a decrease of plasma frequency 𝜔𝑝𝑙 for decreasing of film thickness [36] (𝑡, 𝑡𝐶 < 20 nm) leads to a decrease of permittivity, and this size effects should be taken into account. This reduces the imaginary part, Im[𝜖𝐴𝑢 ] = 𝜔2𝑝𝑙 𝛾𝜔𝑜𝑝𝑡 ∕(𝜔4𝑜𝑝𝑡 + 𝛾 2 𝜔2𝑜𝑝𝑡 ), and increase the real part, 𝜖𝑏 −𝜔2𝑝𝑙 𝜔2 ∕(𝜔4𝑜𝑝𝑡 +𝛾 2 𝜔2𝑜𝑝𝑡 ), of 𝜖𝐴𝑢 as shown by Wang et al. [37]. This effect is due to the discontinuous structure of ultra-thin films, i.e., the film is no longer a continuous medium but it has the topology of heterogeneous clusters that has strong plasmonic effects. Therefore, for ultra-thin films, peaks of the plasma resonance in Fig. 5 will shift to the long-wavelength region. Especially, an increase of the imaginary part simultaneously with a decrease of the real part of the dielectric function 𝜖𝐴𝑢 implies an enhancement of the optical absorption on thin films. And the above Maxwell’s equations cannot be used accurately, i.e., when this plasmonic effect is strong. Here, we use the parameter for film thicknesses which are greater than 20 nm, 𝜖𝑏 = 12, 𝜔𝑝𝑙 = 8.95 eV, and 𝛾 = 72.3 meV. Cantilever length is 233 μm and width is 23 μm. The heat absorption of the back layer (a part of the optical fiber) of the cantilever is assumed to be negligible.
Fig. 5. (Left axis) Thicknesses of the cantilever metallic film 𝑡 (black solid line) and of the coating layer 𝑡𝐶 (red dashed line) to maximize the enhancement ratio 𝜂𝑟 (blue dots, right axis). The dip at 𝜆 ∼ 450 nm corresponds to the absorption band of Au where the field intensity difference between the left and right sides is small. The figure shows that increasing values of 𝑡 > 140 nm or 𝑡𝐶 > 60 nm does not increase 𝜂𝑟 .
wavelength 𝜆, the film thicknesses should be chosen around the range of 𝑡 for the cantilever and 𝑡𝐶 for the coating for a maximal enhancement ratio of RP. In general, the metallic layer on the cantilever (𝑡, black solid line) is ∼30 nm thicker than the fiber-end coating layer (𝑡𝐶 , red dashed line). For example, if the input laser has 𝜆 = 1200 nm, 𝑡 should be ∼55 nm and 𝑡𝐶 ∼ 25 nm, and 𝜂𝑟 reaches 30, a great enhancement of RP. In particular, at the plasmon resonance frequency, where Re[𝜖(𝜔𝑜𝑝𝑡 ) ] = 0, 𝜔2𝑟𝑒𝑠 = 𝜔2𝑝𝑙 ∕𝜖𝑏 − 𝛾 2 , and this leads to 𝜆𝑟𝑒𝑠 = ℎ𝑐∕𝜔𝑟𝑒𝑠 ≃ 450 nm (indicated by the green dotted line), and Maxwell’s equations cannot be solved exactly using the conventional method above; therefore, 𝜂𝑟 cannot be obtained because the optical energy is now transferred to the plasmon oscillations of the electrons. However, approximate values can be deduced, and in Fig. 5 we plot a full range of 𝜂𝑟 that involves 𝜆 < 𝜆𝑟𝑒𝑠 and 𝜆 ≃ 𝜆𝑟𝑒𝑠 to explicitly demonstrate such a resonance effect. The absorbed heat on the cantilever can be given by 𝑑3 =𝑑2 +𝑡
𝑃𝑎𝑏𝑠 =
∫𝑑2
𝑑𝑧|𝐸3 |2 𝐈𝐦[1 − 𝜖𝐴𝑢 ]𝜔𝑜𝑝𝑡 ∕𝑐,
4. Conclusion In summary, we have determined optimal parameters for a system of planar optical multilayers to allow both the radiation pressure and the thermal heating of these cavities to be enhanced greatly. The optimal coating and cantilever thicknesses for two typical input laser wavelengths of 633 nm and 1064 nm have been deduced. A figure for the enhancement factor of the radiation pressure in a wide range of optical wavelength, 300–1500 nm, has been shown. As a result, we obtain a set of parameters for gold thin films in measurement systems such as the fiber-top cantilever. While gold has been used as a typical material to examine the trapping effects, the results that are reported here are universal and are applicable to similar metals such as Ag.
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Acknowledgments
[18] D. Iannuzzi, S. Deladi, J.W. Berenschot, S. de Man, K. Heeck, M.C. Elwenspoek, Fibertop atomic force microscope, Rev. Sci. Instrum. 77 (10) (2006) 106105. http://dx. doi.org/10.1063/1.2358710. URL http://scitation.aip.org/content/aip/journal/rsi/ N.D. Vy is thankful to Associate Professor Ho Trung D. and Professor 77/10/10.1063/1.2358710. L.V. Hoang for their encouragement. This research is funded by Vietnam [19] K.B. Gavan, J.H. Rector, K. Heeck, D. Chavan, G. Gruca, T.H. Oosterkamp, National Foundation for Science and Technology Development (NAFOSD. Iannuzzi, Top-down approach to fiber-top cantilevers, Opt. Lett. 36 (15) (2011) 2898–2900. http://dx.doi.org/10.1364/OL.36.002898. URL http://ol.osa. TED) under grant number 103.01-2016.70. org/abstract.cfm?URI=ol-36-15-2898. [20] M. Cui, C.H. Van Hoorn, D. Ianuuzzi, Miniaturized fibre-top cantilevers on etched References fibres, J. Micros. 264 (3) (2016) 370–374. http://dx.doi.org/10.1111/jmi.12452. [21] J. Li, F. Albri, J.N. Sun, M.M. Miliar, R.R.J. Maier, D.P. Hand, W.N. MacPherson, [1] L. Liu, S. Peng, W. Wen, P. Sheng, Micro thermoindicators and optical-electronic Fabricating optical fibre-top cantilevers for temperature sensing, Meas. Sci. Tech. temperature control for microfluidic applications, Appl. Phys. Lett. 91 (9) (2007) 25 (3) (2014) 035206. URL http://stacks.iop.org/0957-0233/25/i=3/a=035206. 093513. http://dx.doi.org/10.1063/1.2776848. [22] D. Rugar, H.J. Mamin, R. Erlandsson, J.E. Stern, B.D. Terris, Force microscope using [2] C. Wu, H.Y. Fu, K.K. Qureshi, B.-O. Guan, H.Y. Tam, High-pressure and higha fiberoptic displacement sensor, Rev. Sci. Instrum. 59 (11) (1988) 2337–2340. temperature characteristics of a Fabry–Perot interferometer based on photonic http://dx.doi.org/10.1063/1.1139958. crystal fiber, Opt. Lett. 36 (3) (2011) 412–414. http://dx.doi.org/10.1364/OL.36. [23] D. Rugar, H.J. Mamin, P. Guethner, Improved fiberoptic interferometer for atomic 000412. URL http://ol.osa.org/abstract.cfm?URI=ol-36-3-412. force microscopy, Appl. Phys. Lett. 55 (25) (1989) 2588–2590. http://dx.doi.org/ [3] X. Zhang, W. Peng, Temperature-independent fiber salinity sensor based on Fabry10.1063/1.101987. Perot interference, Opt. Express 23 (8) (2015) 10353–10358. http://dx.doi.org/ [24] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, 10.1364/OE.23.010353. URL http://www.opticsexpress.org/abstract.cfm?URI=oeInterference and Diffraction of Light, Cambridge University Press, Cambridge, UK, 23-8-10353. 1997. URL https://books.google.com.vn/books/about/Principles_of_Optics.html?id [4] N.H. Wan, F. Meng, T. Schröder, R.-J. Shiue, E.H. Chen, D. Englund, High-resolution =oV80AAAAIAAJ&redir_esc=y. optical spectroscopy using multimode interference in a compact tapered fibre, [25] H. MacLeod, Thin-Film Optical Filters, third ed., in: Series in Optics and OptoelecNature Commun. 6 (2015) 7762. http://dx.doi.org/10.1038/ncomms8762. tronics, CRC Press, 2001. URL https://books.google.com.vn/books?id=D0S9hxzPJq8C. [5] J. Li, F. Albri, R.R.J. Maier, W. Shu, J. Sun, D.P. Hand, W.N. MacPherson, A micro[26] N. Duy Vy, T. Iida, Ambient-dependent optomechanical control of cantilever with machined optical fiber cantilever as a miniaturized ph sensor, IEEE Sens. J. 15 (12) mechanical nonlinearity by cavity-induced radiation force, Appl. Phys. Lett. 102 (9) (2015) 7221–7228. http://dx.doi.org/10.1109/JSEN.2015.2472971. (2013) 091101. URL http://scitation.aip.org/content/aip/journal/apl/102/9/10. [6] Y. Liu, D.N. Wang, W.P. Chen, Crescent shaped Fabry-Perot fiber cavity for ultra1063/1.4794060. sensitive strain measurement, Sci. Rep. 6 (2016) 38390. http://dx.doi.org/10.1038/ [27] N.D. Vy, L. Tri Dat, T. Iida, Cancelation of thermally induced frequency shifts in srep38390. URL http://www.nature.com/articles/srep38390. bimaterial cantilevers by nonlinear optomechanical interactions, Appl. Phys. Lett. [7] A. Kersey, D. Jackson, M. Corke, A simple fibre Fabry-Perot sensor, Opt. Commun. 109 (5) (2016). http://dx.doi.org/10.1063/1.4960380. URL http://scitation.aip. 45 (2) (1983) 71–74. http://dx.doi.org/10.1016/0030-4018(83)90047-0. URL http: org/content/aip/journal/apl/109/5/10.1063/1.4960380. //www.sciencedirect.com/science/article/pii/0030401883900470. [28] C.M. Hoang, N.D. Vy, L.T. Dat, T. Iida, Enhancing amplitudes of higher-order [8] J.L. Santos, A.P. Leite, D.A. Jackson, Optical fiber sensing with a low-finesse Fabry– eigenmodes of atomic force microscope cantilevers by laser for better mass sensing, Perot cavity, Appl. Opt. 31 (34) (1992) 7361–7366. http://dx.doi.org/10.1364/AO. Japan. J. Appl. Phys. 56 (6S1) (2017) 06GK05. URL http://stacks.iop.org/134731.007361. URL http://ao.osa.org/abstract.cfm?URI=ao-31-34-7361. 4065/56/i=6S1/a=06GK05. [9] T. Wang, S. Zheng, Z. Yang, A high precision displacement sensor using a low[29] M. Bordag, U. Mohideen, V. Mostepanenko, New developments in the Casimir effect, finesse fiber-optic Fabry-Pérot interferometer, Sens. Actuators A 69 (2) (1998) Phys. Rep. 353 (13) (2001) 1–205. http://dx.doi.org/10.1016/S0370-1573(01) 134–138. http://dx.doi.org/10.1016/S0924-4247(98)00075-2. URL http://www. 00015-1. URL http://www.sciencedirect.com/science/article/pii/S0370157301000 sciencedirect.com/science/article/pii/S0924424798000752. 151. [10] L. Perret, L. Chassagne, S. Topu¸, P. Ruaux, B. Cagneau, Y. Alayli, Fiber optics sensor [30] C. Genet, A. Lambrecht, P.M. Neto, S. Reynaud, The casimir force between rough for sub-nanometric displacement and wide bandwidth systems, Sens. Actuators A metallic plates, Europhys. Lett. 62 (4) (2003) 484. http://stacks.iop.org/0295-5075/ 165 (2) (2011) 189–193. http://dx.doi.org/10.1016/j.sna.2010.10.010. URL http: 62/i=4/a=484. //www.sciencedirect.com/science/article/pii/S0924424710004565. [31] V.A. Parsegian, Van der Waals Forces, Cambridge University Press, 2005. URL [11] C.H. Metzger, K. Karrai, Cavity cooling of a microlever, Nature 432 (2004) 1002. https://books.google.com.vn/books?id=K4F7uHJVEOoC. http://dx.doi.org/10.1038/nature03118. URL https://www.nature.com/nature/journal/ [32] M. Bordag, I.V. Fialkovsky, D.M. Gitman, D.V. Vassilevich, Casimir interaction v432/n7020/full/nature03118.html. between a perfect conductor and graphene described by the dirac model, Phys. [12] F. Marquardt, S.M. Girvin, Optomechanics, Physics 2 (2009) 40. http://dx.doi.org/ Rev. B 80 (2009) 245406. http://dx.doi.org/10.1103/PhysRevB.80.245406. URL 10.1103/Physics.2.40. URL http://link.aps.org/doi/10.1103/Physics.2.40. https://link.aps.org/doi/10.1103/PhysRevB.80.245406. [13] B.I. Kim, L. Smith, T. Tran, S. Rossland, E. Parkinson, Cantilever-based optical [33] T. Iida, H. Ishihara, Study of the mechanical interaction between an electromagnetic interfacial force microscope in liquid using an optical-fiber tip, AIP Adv. 3 (3) (2013) field and a nanoscopic thin film near electronic resonance, Opt. Lett. 27 (2002) 754. 032126. http://dx.doi.org/10.1063/1.4795761. http://dx.doi.org/10.1364/OL.27.000754. [14] H.Y. Choi, K.S. Park, S.J. Park, U.-C. Paek, B.H. Lee, E.S. Choi, Miniature fiber-optic [34] A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: high temperature sensor based on a hybrid structured Fabry–Perot interferometer, A Reprint Volume with Commentaries, 2006. URL https://books.google.com.vn/ Opt. Lett. 33 (21) (2008) 2455–2457. http://dx.doi.org/10.1364/OL.33.002455. books?id=VGzVCgAAQBAJ. URL http://ol.osa.org/abstract.cfm?URI=ol-33-21-2455. [35] N.D. Vy, T. Iida, Enhancing thermally induced effects on atomic force microscope [15] N.D. Vy, T. Iida, Nonlinear analysis of sub-millikelvin optomechanical cooling for cantilevers using optical microcavities, Appl. Phys. Express 9 (12) (2016) 126601. extremely low noise quantum measurement, Appl. Phys. Express 8 (3) (2015) URL http://stacks.iop.org/1882-0786/9/i=12/a=126601. 032801. URL http://stacks.iop.org/1882-0786/8/i=3/a=032801. [36] S. Laref, J. Cao, A. Asaduzzaman, K. Runge, P. Deymier, R.W. Ziolkowski, [16] D. Iannuzzi, S. Deladi, V.J. Gadgil, R.G.P. Sanders, H. Schreuders, M.C. Elwenspoek, M. Miyawaki, K. Muralidharan, Size-dependent permittivity and intrinsic optical Monolithic fiber-top sensor for critical environments and standard applications, anisotropy of nanometric gold thin films: a density functional theory study, Opt. Appl. Phys. Lett. 88 (5) (2006) 053501. http://dx.doi.org/10.1063/1.2170139. URL Express 21 (10) (2013) 11827–11838. http://dx.doi.org/10.1364/OE.21.011827. http://scitation.aip.org/content/aip/journal/apl/88/5/10.1063/1.2170139. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-21-10-11827. [17] S. Deladi, D. Iannuzzi, V.J. Gadgil, H. Schreuders, M.C. Elwenspoek, Carving fiber[37] X. Wang, K. ping Chen, M. Zhao, D.D. Nolte, Refractive index and dielectric top optomechanical transducers from an optical fiber, J. Micromech. Microeng. constant evolution of ultra-thin gold from clusters to films, Opt. Express 18 (24) 16 (5) (2006) 886. URL http://stacks.iop.org/0960-1317/16/i=5/a=002. (2010) 24859–24867. http://dx.doi.org/10.1364/OE.18.024859. URL http://www. opticsexpress.org/abstract.cfm?URI=oe-18-24-24859.
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